```Date: Mar 20, 2000 10:39 PM
Author: eppstein@euclid.ics.uci.edu
Subject: Re: unitary (Egyptian) fractions

haoyuep@aol.com (Dan Hoey) writes:> >   What is the earliest such rational that cannot be expressed as the> > sum of n unit fractions?> I'm sure you mean n _distinct_ unit fractions.If it can be expressed as n unit fractions it can be expressed as ndistinct unit fractions by repeatedly using the transformation	1/(2k-1) + 1/(2k-1)  ==>  1/k + 1/k(2k-1)> By the way, I found that 732/733 has 2771 different seven-term> representations.  The largest denominator appears in the> representation (2305193137933140 33397845 4484 45 7 3 2).  The> smallest maximum denominator appears in (26388 20524 7330 45 7 3 2).Damn, you beat me to it.  Just to solve this problem, I modified theSmallMultiples method in my EgyptianFraction Mathematica routines(http://www.ics.uci.edu/~eppstein/numth/egypt/Egypt.m) to use dynamicprogramming instead of brute force, so now it's not hopeless to solve itthat way but it's still slow.I posted a couple days ago the min-max denominator of any representation(7330, with a ten-term representation) and the best denominator for anine-term representation (8063).  I am now also able to find the besteight-term representation:732/733 = 1/2 + 1/3 + 1/9 + 1/20 + 1/255 + 1/8796 + 1/12461 + 1/13194-- David Eppstein       UC Irvine Dept. of Information & Computer Scienceeppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/
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